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          模幂运算
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        <p>&ensp;&ensp;&ensp;&ensp;由于格式原因可能文章显示有问题，a^1234^代表a的1234次幂</p>
<p>&ensp;&ensp;&ensp;&ensp;a^1234^  =  a^4^  *  (a^123^)^10^</p>
<p>&ensp;&ensp;&ensp;&ensp;(a * b) % mod = (a % mod)(b % mod) % mod</p>
<span id="more"></span>

<h4 id="50-Pow-x-n"><a href="#50-Pow-x-n" class="headerlink" title="50. Pow(x, n)"></a><a target="_blank" rel="noopener" href="https://leetcode-cn.com/problems/powx-n/">50. Pow(x, n)</a></h4><p>&ensp;&ensp;&ensp;&ensp;实现 pow(<em>x</em>, <em>n</em>) ，即计算 x 的 n 次幂函数（即，x^n^）。</p>
<p><strong>示例 1：</strong></p>
<figure class="highlight plain"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br></pre></td><td class="code"><pre><span class="line">输入：x &#x3D; 2.00000, n &#x3D; 10</span><br><span class="line">输出：1024.00000</span><br></pre></td></tr></table></figure>

<p><strong>示例 2：</strong></p>
<figure class="highlight plain"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br></pre></td><td class="code"><pre><span class="line">输入：x &#x3D; 2.10000, n &#x3D; 3</span><br><span class="line">输出：9.26100</span><br></pre></td></tr></table></figure>

<p><strong>示例 3：</strong></p>
<figure class="highlight plain"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br></pre></td><td class="code"><pre><span class="line">输入：x &#x3D; 2.00000, n &#x3D; -2</span><br><span class="line">输出：0.25000</span><br><span class="line">解释：2^(-2) &#x3D; 1&#x2F;22 &#x3D; 1&#x2F;4 &#x3D; 0.25</span><br></pre></td></tr></table></figure>

<p><strong>提示：</strong></p>
<ul>
<li>  <code>-100.0 &lt; x &lt; 100.0</code></li>
<li>  <code>-231 &lt;= n &lt;= 231-1</code></li>
<li>  <code>-10^4 &lt;= x^n &lt;= 10^4</code></li>
</ul>
<p>&ensp;&ensp;&ensp;&ensp;递归：</p>
<figure class="highlight cpp"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br></pre></td><td class="code"><pre><span class="line"><span class="class"><span class="keyword">class</span> <span class="title">Solution</span> </span></span><br><span class="line"><span class="class">&#123;</span></span><br><span class="line"><span class="keyword">public</span>:</span><br><span class="line">    <span class="function"><span class="keyword">double</span> <span class="title">quickMul</span><span class="params">(<span class="keyword">double</span> x, <span class="keyword">long</span> <span class="keyword">long</span> N)</span> </span></span><br><span class="line"><span class="function">    </span>&#123;</span><br><span class="line">        <span class="keyword">if</span> (N == <span class="number">0</span>) </span><br><span class="line">        &#123;</span><br><span class="line">            <span class="keyword">return</span> <span class="number">1.0</span>;</span><br><span class="line">        &#125;</span><br><span class="line">        <span class="keyword">double</span> y = <span class="built_in">quickMul</span>(x, N / <span class="number">2</span>);</span><br><span class="line">        <span class="keyword">return</span> N % <span class="number">2</span> == <span class="number">0</span> ? y * y : y * y * x;</span><br><span class="line">    &#125;</span><br><span class="line"></span><br><span class="line">    <span class="function"><span class="keyword">double</span> <span class="title">myPow</span><span class="params">(<span class="keyword">double</span> x, <span class="keyword">int</span> n)</span> </span></span><br><span class="line"><span class="function">    </span>&#123;</span><br><span class="line">        <span class="keyword">long</span> <span class="keyword">long</span> N = n;</span><br><span class="line">        <span class="keyword">return</span> N &gt;= <span class="number">0</span> ? <span class="built_in">quickMul</span>(x, N) : <span class="number">1.0</span> / <span class="built_in">quickMul</span>(x, -N);</span><br><span class="line">    &#125;</span><br><span class="line">&#125;;</span><br></pre></td></tr></table></figure>

<p>&ensp;&ensp;&ensp;&ensp;迭代：</p>
<figure class="highlight cpp"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br></pre></td><td class="code"><pre><span class="line"><span class="class"><span class="keyword">class</span> <span class="title">Solution</span> &#123;</span></span><br><span class="line"><span class="keyword">public</span>:</span><br><span class="line">    <span class="function"><span class="keyword">double</span> <span class="title">quickMul</span><span class="params">(<span class="keyword">double</span> x, <span class="keyword">long</span> <span class="keyword">long</span> N)</span> </span></span><br><span class="line"><span class="function">    </span>&#123;</span><br><span class="line">        <span class="keyword">double</span> ans = <span class="number">1.0</span>;</span><br><span class="line">        <span class="comment">// 贡献的初始值为 x</span></span><br><span class="line">        <span class="keyword">double</span> x_contribute = x;</span><br><span class="line">        <span class="comment">// 在对 N 进行二进制拆分的同时计算答案</span></span><br><span class="line">        <span class="keyword">while</span> (N &gt; <span class="number">0</span>)</span><br><span class="line">        &#123;</span><br><span class="line">            <span class="keyword">if</span> (N &amp; <span class="number">1</span>) </span><br><span class="line">            &#123;</span><br><span class="line">                <span class="comment">// 如果 N 二进制表示的最低位为 1，那么需要计入贡献</span></span><br><span class="line">                ans *= x_contribute;</span><br><span class="line">            &#125;</span><br><span class="line">            <span class="comment">// 将贡献不断地平方</span></span><br><span class="line">            x_contribute *= x_contribute;</span><br><span class="line">            <span class="comment">// 舍弃 N 二进制表示的最低位，这样我们每次只要判断最低位即可</span></span><br><span class="line">            N /= <span class="number">2</span>;</span><br><span class="line">        &#125;</span><br><span class="line">        <span class="keyword">return</span> ans;</span><br><span class="line">    &#125;</span><br><span class="line"></span><br><span class="line">    <span class="function"><span class="keyword">double</span> <span class="title">myPow</span><span class="params">(<span class="keyword">double</span> x, <span class="keyword">int</span> n)</span> </span></span><br><span class="line"><span class="function">    </span>&#123;</span><br><span class="line">        <span class="keyword">long</span> <span class="keyword">long</span> N = n;</span><br><span class="line">        <span class="keyword">return</span> N &gt;= <span class="number">0</span> ? <span class="built_in">quickMul</span>(x, N) : <span class="number">1.0</span> / <span class="built_in">quickMul</span>(x, -N);</span><br><span class="line">    &#125;</span><br><span class="line">&#125;;</span><br><span class="line"></span><br></pre></td></tr></table></figure>



<p>&ensp;&ensp;&ensp;&ensp;a^1234^  =  a^4^  *  (a^123^)^10^  $\Longrightarrow$ a^[1,2,3,4]^  =  a^4^  *  (a^[1,2,3]^)^10^ </p>
<h4 id="372-超级次方"><a href="#372-超级次方" class="headerlink" title="372. 超级次方"></a><a target="_blank" rel="noopener" href="https://leetcode-cn.com/problems/super-pow/">372. 超级次方</a></h4><p>你的任务是计算  a^b^  对 <code>1337</code> 取模，<code>a</code> 是一个正整数，<code>b</code> 是一个非常大的正整数且会以数组形式给出。</p>
<p><strong>示例 1：</strong></p>
<figure class="highlight plain"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br></pre></td><td class="code"><pre><span class="line">输入：a &#x3D; 2, b &#x3D; [3]</span><br><span class="line">输出：8</span><br></pre></td></tr></table></figure>

<p><strong>示例 2：</strong></p>
<figure class="highlight plain"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br></pre></td><td class="code"><pre><span class="line">输入：a &#x3D; 2, b &#x3D; [1,0]</span><br><span class="line">输出：1024</span><br></pre></td></tr></table></figure>

<p><strong>示例 3：</strong></p>
<figure class="highlight plain"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br></pre></td><td class="code"><pre><span class="line">输入：a &#x3D; 1, b &#x3D; [4,3,3,8,5,2]</span><br><span class="line">输出：1</span><br></pre></td></tr></table></figure>

<p><strong>示例 4：</strong></p>
<figure class="highlight plain"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br></pre></td><td class="code"><pre><span class="line">输入：a &#x3D; 2147483647, b &#x3D; [2,0,0]</span><br><span class="line">输出：1198</span><br></pre></td></tr></table></figure>

<p> <strong>提示：</strong></p>
<ul>
<li>  <code>1 &lt;= a &lt;= 231 - 1</code></li>
<li>  <code>1 &lt;= b.length &lt;= 2000</code></li>
<li>  <code>0 &lt;= b[i] &lt;= 9</code></li>
<li>  <code>b</code> 不含前导 0</li>
</ul>
<figure class="highlight cpp"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br></pre></td><td class="code"><pre><span class="line"><span class="class"><span class="keyword">class</span> <span class="title">Solution</span> </span></span><br><span class="line"><span class="class">&#123;</span></span><br><span class="line"><span class="keyword">public</span>:</span><br><span class="line">    <span class="keyword">const</span> <span class="keyword">int</span> mod = <span class="number">1337</span>;</span><br><span class="line"></span><br><span class="line">    <span class="function"><span class="keyword">int</span> <span class="title">myPow</span> <span class="params">(<span class="keyword">int</span> x, <span class="keyword">int</span> n)</span></span></span><br><span class="line"><span class="function">    </span>&#123;</span><br><span class="line">        x %= mod;</span><br><span class="line">        <span class="keyword">int</span> res = <span class="number">1</span>;</span><br><span class="line">        <span class="keyword">while</span> (n) </span><br><span class="line">        &#123;</span><br><span class="line">            <span class="keyword">if</span> (n &amp; <span class="number">1</span>) res = res * x % mod;</span><br><span class="line">            x = x * x % mod;</span><br><span class="line">            n &gt;&gt;= <span class="number">1</span>;</span><br><span class="line">        &#125;</span><br><span class="line">        <span class="keyword">return</span> res;</span><br><span class="line">    &#125;</span><br><span class="line"></span><br><span class="line">    <span class="function"><span class="keyword">int</span> <span class="title">superPow</span><span class="params">(<span class="keyword">int</span> a, vector&lt;<span class="keyword">int</span>&gt;&amp; b)</span> </span></span><br><span class="line"><span class="function">    </span>&#123;</span><br><span class="line">        <span class="keyword">if</span> (b.<span class="built_in">empty</span>())</span><br><span class="line">        &#123;</span><br><span class="line">            <span class="keyword">return</span> <span class="number">1</span>;</span><br><span class="line">        &#125;</span><br><span class="line">        <span class="keyword">int</span> lastValue = b.<span class="built_in">back</span>();</span><br><span class="line">        b.<span class="built_in">pop_back</span>();</span><br><span class="line"></span><br><span class="line">        <span class="keyword">int</span> part1 = <span class="built_in">myPow</span>(a, lastValue);</span><br><span class="line">        <span class="keyword">int</span> part2 = <span class="built_in">myPow</span>(<span class="built_in">superPow</span>(a, b), <span class="number">10</span>);</span><br><span class="line">        <span class="keyword">return</span> part1 * part2 % mod;  </span><br><span class="line">    &#125;</span><br><span class="line">&#125;;</span><br></pre></td></tr></table></figure>

<p><a target="_blank" rel="noopener" href="https://mp.weixin.qq.com/s?__biz=MzAxODQxMDM0Mw==&mid=2247485035&idx=1&sn=c03c9056f412bf590480156e4357b433&chksm=9bd7f863aca07175eca0bd5e638c857969af1936cc2b7b4a01610809ba1d85a3272703e2b3a2&scene=21#wechat_redirect">Super Pow：如何高效进行模幂运算 (qq.com)</a></p>
<p><a target="_blank" rel="noopener" href="https://mp.weixin.qq.com/s?__biz=MzA5MTM1NTc2Ng==&mid=2458321758&idx=1&sn=e92e7a72d245831c29acb3d2a29607da&scene=21#wechat_redirect">客户端基本不用的算法系列：快速幂 (qq.com)</a></p>
<p><a target="_blank" rel="noopener" href="https://blog.csdn.net/FlushHip/article/details/82495034">右移一位和除二的区别</a></p>

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